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Ideal lo-pass filter

  1. Oct 27, 2008 #1
    It is well-known that the step-response of an ideal low-pass filter (Gain = 1 for f = 0 to F and = 0 for f = F to infinity) is non-causal, in that the output appeares before t = 0.

    But what if the filter's Gain is = 1 for f = 0 to F and = e (some small non-zero value) for f = F to infinity? Then the Paley-Wiener criterion for realizability is satisfied, and the output must be causal. What then is the output?
     
  2. jcsd
  3. Oct 27, 2008 #2

    rbj

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    with sufficient delay (and you weren't spec-ing the phase response of the nascent ideal LPF), you can have your causal impulse response get closer and closer to a delayed sinc() function.
     
  4. Oct 28, 2008 #3
    Thanks for responding! I think this is fun! It is also, no doubt, well-known, but I haven't seen it anywhere.

    But a sinc() function is never causal, no matter what the delay, and I want a casual output function. There has to be one.

    Let's say the filter phase is zero over the entire band, then what's the output for an inpulse input at t = 0 ?
     
  5. Oct 28, 2008 #4

    rbj

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    i said "closer" to a delayed sinc function. if you delay it enough, the portion of the sinc() function that precedes t=0 has sufficiently low amplitude that the difference between it and zero is small. so, if we can delay the sinc() function sufficiently, we can get as close to a causal sinc() as we want.

    you can't have it zero phase over the entire band and causal at the same time (unless the impulse response was zero for all t <> 0). for it to be zero phase over the entire band, then the impulse response would have to be symmetrical about t=0 (even symmetry).
     
  6. Oct 28, 2008 #5
    Rbj's already answered the substance of this question, but could you state the version of the Paley-Wiener criterion that you have in mind. IIRC, simply making the magnitude response nonzero is not enough to satisfy the criterion.
     
  7. Oct 31, 2008 #6
    I'm only aware of one version of the criterion, namely:

    The N&S condition for a linear-time-invariant filter to have a causal response is that its Gain versus frequency, G(f), should satisfy:

    The integral from -infinity to +infinity of df*|log(G(f)|/(1+f^2) be < infinity.

    If you know of another one, please let me know.

    Thanks for responding.
     
  8. Nov 1, 2008 #7

    rbj

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    hey, bobby, you need to use LaTeX here:

    [tex] \int_{-\infty}^{+\infty} \frac{|\log(G(f)|}{1+f^2} \ df \ \ < \infty [/tex]

    for causality in the time domain, it is necessary and sufficient that the real part and imaginary part of [itex]G(f)[/itex] be a Hilbert Transform pair.

    [tex] \frac{1}{\pi} \int_{-\infty}^{+\infty} \ \frac{\mathrm{Re}\{(G(u)\}}{f-u} \ du = \mathrm{Im}\{(G(f)\} [/tex]
     
  9. Nov 1, 2008 #8
    Okay, so it does seem that a step function

    [tex]
    G(f) = (1-\epsilon)1_{F_{pass}}(f) + \epsilon
    [/tex]

    where [itex]F_{pass}[/itex] is a set of passband frequencies, will satisfy the PW condition:

    [tex]
    \int \frac{|\log G(f)|}{1+f^2}df = 2|\log \epsilon |(\frac{\pi}{2}-\arctan (f_c))<\infty
    [/tex]

    if [itex] G(f)[/itex] is what we might call an "[itex]\epsilon[/itex]-ideal low-pass filter" with cutoff [itex]f_c[/itex] (and real impulse response). This is interesting, because it has an infinitely steep transition band. However, the theorem only tells us that there's a causal filter with this magnitude response. It doesn't tell us what the delay is, or even if the delay is finite. What would the group delay of such a filter look like? Seems like the discontinuity is going to make the delay diverge around [itex]f_c[/itex], no?

    Is there another theorem that relates how large the value of the integral is to the required delay? Something along the lines of [itex]|G(f)|[/itex] being bounded by [itex]Ae^{-b|f|}+c[/itex], IIRC? I'm lacking good sources on this stuff myself, so any references would be appreciated.
     
    Last edited: Nov 1, 2008
  10. Nov 1, 2008 #9
    Look, I'm not a mathematician, as you no doubt must realize (I don't even know how to do LaTeX!) and know nothing about the Paley-Wiener Criterion (although I did attend a lecture by Wiener once!). I saw it in a book by Chester Page and it intrigued me. I don't know what it says about the delay or whether the delay is infinite, nor do I know how to find out. If the delay IS infinite, then my filter doesn't make much sense! If I did see a derivation of this, I probably wouldn't be able to follow it! I'm just fastinated by the subject and I SURE appreciate you guys responding to me about it! Thanks again!
    bobbyk
     
  11. Nov 3, 2008 #10
    The best way to learn [itex]\LaTeX[/itex] is to use the
     
  12. Nov 3, 2008 #11

    rbj

    User Avatar

    EE101?? freshman-level engineering (and math)? i don't think i heard the words "impulse response" until i was a sophomore and started doing diff eqs. on an RC circuit. "infinite support"? that's Advanced Calculus or Real Analysis, ain't it? i s'pose one can easily explain what causality is at the 101 level.
     
  13. Nov 3, 2008 #12
    Where I come from, 101 is a junior-year course. The idea being that you can't really start on stuff in earnest until you finish calculus and physics, which takes two years, I guess..

    Colleges need to get together and settle on some kind of semi-regular course number conventions...
     
  14. Nov 3, 2008 #13

    rbj

    User Avatar

    for my undergrad (U of North Dakota) 1xx was freshman (normally), 2xx sophomore, 3xx junior, 4xx senior, 5xx+ graduate. courses numbered lower than 100 (usually more than 90) meant courses that were remedial in nature and usually were not applicable for credit to the degree. but, if the student was deficient in something, may be required before moving on to a normal college level course. e.g. MATH090 was a high-school level algebra remedial course. there is a general math and science grad requirement for every student and students who flunked some math placement exam (or maybe it was a deficient ACT or SAT score or maybe they found themselves drowning in the first required math course, i dunno) those folks would take something like an 090 course.

    traditionally MATH101, PHYS101, PHIL101, MUS101, ECON101, PSY101, CHEM101, BIO101, or ENGL101 meant the entry-level course in the discipline. we had an ENG101 (in general engineering). the first EE circuits course was EE201 or similarly numbered.

    there are papers (in fact, i wrote one) that are written intended to be a sorta primer or tutorial or somehow seminal paper on a subject. i believe that that practice came from the common understanding of "101" being entry level, but i guess that wouldn't mean freshman level. i s'pose that's how you meant to use the term.

    interesting etymology of a modern term.


    i think you can start doing some circuit analysis before you get done with calc and physics. you need to know what a derivative and integral is. even though it would be nice to have the first two courses in General Physics behind (so you have the physical foundation to KVL, KCL, and the volt-amp characteristics of the R's, L's, and C's), you could start with KVL, KCL, and the volt-amp characteristics as axiomatic, the rules that you begin with. and let the physical justification come later.

    it would be nice if there was some standardization so that credits would transfer easily. i guess that's what organizations like ABET are for.
     
    Last edited: Nov 3, 2008
  15. Nov 3, 2008 #14
    Ah, at my alma mater, 0-99 are lower-division (fresh/soph) courses, 100-199 are upper-division (junior/senior) and 200-299 are graduate.

    Yeah, I suppose I was thinking of linear systems theory as "ECE101" (which it happens literally to be where I went to school), but on second thought this might simply reflect my biases as someone in the signals/systems end of the field. Most people would probably consider circuit analysis to be "101" material here.

    Either way, I certainly wouldn't expect a "101" course of any type (be it EE, or a more specific signals/systems theory course) to actually get into the details of PW and spectral factorization. That's grad school stuff where I come from. I was thinking more of the lazy "ideal lowpass = noncausal impulse response" explanation that most undergrads leave school with. The more I think about it, the more it seems to me that you can get around non-causality, but not without incurring infinite delay. It's the transition width that's the real issue.

    Oh, you definitely can. It's just that you have to re-do it all the next year once you know calculus. Mostly they limit it to steady-state response of linear circuits with very simple driving functions (just one or two sinusoids, say, or even just DC), in which case all that's needed is a little bit of familiarity with complex numbers.
     
  16. Jan 2, 2009 #15
    It is my understanding that if the Paley-Wiener criterion for the GAIN is satisfied, then there
    is a PHASE associated with that GAIN such that the impulse response is casual and has ZERO delay. How do I find that PHASE?

    Thans for youir interest!

    bobbyk


     
    Last edited by a moderator: Jan 5, 2009
  17. Jan 5, 2009 #16
    Err, zero delay? That doesn't sound right; what's the definition of "delay" here? I don't see how a nontrivial filter can be both causal and have zero delay...
     
  18. Jan 6, 2009 #17
    qradraphonics: Thanks for your response and your interest!

    I'm sorry for using an undefined term such as "delay", but what is your definition
    of "nontrivial" ?
    I'm sure you know that there are causal filters whose impulse response to an impulse at t = 0 contain an impulse at t = 0. I would regard this as "zero delay". Maybe these are
    "trivial" filters?

    bobbyk
     
  19. Jan 7, 2009 #18
    Yes, that's exactly what I had in mind when I said "trivial filters." That system has a flat frequency response, so it's certainly not the case that you can construct a zero-delay filter with some arbitrary (PW-satisfying) magnitude response. Any causal filter with a non-flat magnitude response is going to have some kind of delay, at least under the definitions of "delay" that I'm familiar with.

    I would guess that the definition of "delay" that your statement holds under would be "phase response is piecewise constant," i.e., it's only a set of measure 0 that contains all the delay?
     
  20. Jan 8, 2009 #19
    There must be a misunderstanding here, as a simple filter having a capacitor from input to output and a resistor from output to ground has, for an impulse input at t=0, an output containing an impulse at t=0. This is what I call zero-delay and the gain is certainly NOT flat.
     
  21. Jan 9, 2009 #20
    That is true, but that only implies zero-delay if by that you mean "the impulse response contains an impulse at t=0." Which is not an entirely unreasonable definition, but we should note that if you input into such a system a tone burst with center frequency around the cut-off frequency of this filter (i.e., 1/RC), then the resulting output tone burst will display a non-zero delay relative to the input.

    Any time the phase response is not a constant for all frequencies, it implies that there are some sets of frequencies that will exhibit delay, and so it seems problematic to call any such filter "zero delay."

    When considering definitions of delay defined directly in terms of the impulse response, it's more usual to use something like the median or mean of the impulse response, rather than the mode (as you've done above). Using either of those definitions will assign the first-order highpass system a non-zero delay, as there is mass to the right of t=0, but not to the left.

    Thinking about it a bit more, the only examples I can think of where delay is defined in terms of the mode (or "peak") of the impulse response is in the context of linear phase systems, where the symmetry constraint implies that the mode is equal to the (constant) group delay.
     
    Last edited: Jan 10, 2009
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